Let \(\mathbf {d}_n= (\mathbf {d}_n^-,\mathbf {d}_n^+)=((d_1^-,d_1^+),\ldots ,(d_n^-,d_n^+))\)  be a directed degree sequence on \(n\)  vertices and let \(m_n = \sum d_i^- = \sum d_i^+\) .
The directed configuration model on \(\mathbf {d}_n\) , introduced by Cooper and Frieze [1]}, is the random directed multigraph on \([n]\)  obtained by associating with vertex \(i\)  \(d_i^-\)  in-stubs and \(d_i^+\)  in-stubs, and then choosing a perfect matching of in- and out- stubs uniformly at random.
We will denote this model by \(G(\mathbf {d}_n)\) .
This is a directed generalisation of the configuration model of Bollobás [2]} which since its introduction has become one of the most widely used random graph models.
